2003/07/16 by Sara Billey, Victor Guillemin, Billey, Sara +3
Mathematics · #05E15 (Primary) #52B20 #53D20 #68W30 (Secondary) #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.CO #math.RT #math.SG #msc:05E15 #msc:52B20 #msc:53D20 #msc:68W30
paper · pdf · doi:10.48550/arxiv.math/0307227
34 pages, 11 figures and diagrams; submitted to Journal of Algebra
arxiv created 2003/07/16 · openalex publication_date 2003/07/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra slk(C) (type Ak-1) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat-Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl4(C) (A3).